Positive dependence and monotonicity in conditional distributions

Positively dependent random variables exhibit the property that an extreme value of one of the variables tends to be accompanied by extreme values of the others. In this paper we define two notions of positive dependence which lead to monotonicity theorems for conditional distributions.     

Distribution-free tests for independence against positive quadrant dependence: A generalization

A class of distribution-free tests based on two matched pairs is considered for testing independence against positive quadrant dependence. The class of tests proposed by Kochar and Gupta (1990) is a member of the proposed class. The performance of the proposed class is evaluated in terms of Pitman asymptotic relative efficiency for Block-Basu (1974) model and Woodworth family of distributions. The small sample performance of few members of the proposed class is also studied by finding empirical powers. Unbiasedness and consistency of the proposed class of tests have been established.     

A note on positive dependence im multivariate distributioms

I counterexample is presented to the result by Alam and Malleolus asserting that if Y is stochastically increasing in a random vector X, then Y is stochastically increasing in a subvector of X. Their result concerning m*-positive dependence, whose proof relies on the erroneous result, is still true.     

“Weak” stochastic orderings and dependence measures

Many notions of dependence rely upon orderings of random pairs. These orderings are generally partial orders, and thus there are many pairs of random vectors which are not comparable. By using a weakened version of stochastic dominance, many new orderings, as well as corresponding dependence measures, are created. The application to stock market data is explored.     

On the Characterization of Copulas by Differential Equations

We study the semigroup action induced by univariate conditioning of copulas. Based on this, we give a new characterization of bivariate copulas in terms of flows generated by solutions of ordinary differential equations with not necessary continuous right side. Several applications, related to concordance ordering of copulas, illustrate the usefulness of this result.     

Some concepts of mult1variate dependence

Several types of positive dependence are shown to be equiv-alent to the concept of a distribution with a density which is TP₂ in pairs. Among these is the concept of m^?-positive depend-ence of Alam and Wallenius. Using this result, all relationships among many of the most important concepts of positive dependence are determined. Furthermore, an application of the equivalences of these types of positive dependence yields a results of Ahmed, Langberg, Leon and Proschan.     

Projection estimators of Pickands dependence functions

The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme‐value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite‐dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite‐dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one.     

A kolmogorov‐smirnov type test for positive quadrant dependence

The author considers a consistent, Kolmogorov‐Smirnov type of test of the complete set of restrictions that relate to the copula representation of positive quadrant dependence. For such a test, he proposes and justifies inference relying on a simulation‐based multiplier method and a bootstrap method. He also explores the finite‐sample behaviour of both methods with Monte Carlo experiments. A first empirical illustration is given for American insurance claim data. A second one examines the presence of positive quadrant dependence in life expectancies at birth of males and females across countries.     

Positive quadrant dependence testing and constrained copula estimation

Positive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution of the test statistics under the null hypothesis of positive quadrant dependence is to resample from a constrained copula. This requires constrained estimation of a copula function. We show that this use of resampling under a constrained copula improves considerably the power performance of existing testing procedures. We propose two resampling procedures, one based on a parametric constrained copula estimation and one relying on nonparametric estimation of a positive quadrant dependence copula, and discuss their properties. The finite‐sample performances of the resulting testing procedures are evaluated via a simulation study that also includes comparisons with existing tests. Finally, a data set of Danish fire insurance claims is tested for positive quadrant dependence. The Canadian Journal of Statistics 41: 36–64; 2013 © 2012 Statistical Society of Canada     

Tail dependence for skew Laplace distribution and skew Cauchy distribution

ABSTRACT

Coefficient of tail dependence measures the strength of dependence in the tail of a bivariate distribution and it has been found useful in the risk management. In this paper, we derive the upper tail dependence coefficient for a random vector following the skew Laplace distribution and the skew Cauchy distribution, respectively. The result shows that skew Laplace distribution is asymptotically independent in upper tail, however, skew Cauchy distribution has asymptotic upper tail dependence.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

André Dabrowski's work on limit theorems and weak dependence

André Robert Dabrowski, Professor of Mathematics and Dean of the Faculty of Sciences at the University of Ottawa, died October 7, 2006, after a short battle with cancer. The author of the present paper, a long‐term friend and collaborator of André Dabrowski, gives a survey of André's work on weak dependence and limit theorems in probability theory. The Canadian Journal of Statistics 37: 307–326; 2009 © 2009 Statistical Society of Canada